CONVERGENCE RATE OF ESTIMATES OF PROBABILITY DENSITY FUNCTION,ITS DERIVATIVES AND MODE

Under various conditions imposed on the kernel k（x） and density f（x）, the rates of thekernel estimate f（x） converging uniformly to f（x） are studied. For example, if f is provedto satisfy the Lipschitz condition of λ-th order for some λ∈[0, 1], we havesup|f（x） - f（x）| = o(（logn/n）loglogn), a. s.Also, the rates of convergence of f（x）→f（x） a. s. and those of the estimate θof thepopulation mode θ are considered.